An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (2)
Calculate the paired differences: d i = x 2 i − x 1 i .
Calculate the mean of the paired differences: d ˉ = − 4.1 .
Calculate the 99% confidence interval: ( − 24.4 , 16.2 ) .
The 99% confidence interval for the true difference between the population means is ( − 24.4 , 16.2 ) .
Explanation
Calculate Paired Differences First, we need to calculate the paired differences d i = x 2 i − x 1 i for each voter. Here's how we find these differences:
d 1 = 68 − 93 = − 25 d 2 = 60 − 64 = − 4 d 3 = 71 − 43 = 28 d 4 = 72 − 69 = 3 d 5 = 53 − 58 = − 5 d 6 = 81 − 51 = 30 d 7 = 66 − 73 = − 7 d 8 = 67 − 88 = − 21 d 9 = 43 − 59 = − 16 d 10 = 64 − 88 = − 24
So, the paired differences are: − 25 , − 4 , 28 , 3 , − 5 , 30 , − 7 , − 21 , − 16 , − 24 .
Calculate the Mean of Paired Differences Next, we calculate the sample mean of the paired differences, denoted as d ˉ . This is the sum of the differences divided by the number of pairs (which is 10 in this case): d ˉ = 10 ∑ i = 1 10 d i = 10 − 25 − 4 + 28 + 3 − 5 + 30 − 7 − 21 − 16 − 24 = 10 − 41 = − 4.1 So, the sample mean of the paired differences is − 4.1 .
Calculate the Standard Deviation of Paired Differences Now, we need to calculate the sample standard deviation of the paired differences, denoted as s d . This measures the spread of the differences around the mean difference. First, we calculate the squared differences from the mean, then sum them, divide by n − 1 , and take the square root:
s d = 10 − 1 ∑ i = 1 10 ( d i − d ˉ ) 2
Let's compute this step by step:
∑ i = 1 10 ( d i − d ˉ ) 2 = ( − 25 − ( − 4.1 ) ) 2 + ( − 4 − ( − 4.1 ) ) 2 + ( 28 − ( − 4.1 ) ) 2 + ( 3 − ( − 4.1 ) ) 2 + ( − 5 − ( − 4.1 ) ) 2 + ( 30 − ( − 4.1 ) ) 2 + ( − 7 − ( − 4.1 ) ) 2 + ( − 21 − ( − 4.1 ) ) 2 + ( − 16 − ( − 4.1 ) ) 2 + ( − 24 − ( − 4.1 ) ) 2
= ( − 20.9 ) 2 + ( − 0.1 ) 2 + ( 32.1 ) 2 + ( 7.1 ) 2 + ( − 0.9 ) 2 + ( 34.1 ) 2 + ( − 2.9 ) 2 + ( − 16.9 ) 2 + ( − 11.9 ) 2 + ( − 19.9 ) 2
= 436.81 + 0.01 + 1030.41 + 50.41 + 0.81 + 1162.81 + 8.41 + 285.61 + 141.61 + 396.01
= 3512.9
Now, divide by n − 1 = 10 − 1 = 9 :
9 3512.9 = 390.3222
Finally, take the square root:
s d = 390.3222 ≈ 19.7566
So, the sample standard deviation of the paired differences is approximately 19.8 .
Find the Critical t-value Now we need to find the critical t-value for a 99% confidence interval with n − 1 = 10 − 1 = 9 degrees of freedom. For a 99% confidence level, α = 1 − 0.99 = 0.01 , so α /2 = 0.005 . Looking up t 0.005 , 9 in a t-table (or using a calculator), we find that t 0.005 , 9 ≈ 3.250 .
Calculate the Margin of Error Next, we calculate the margin of error E using the formula:
E = t α /2 ⋅ n s d = 3.250 ⋅ 10 19.7566 ≈ 3.250 ⋅ 3.1623 19.7566 ≈ 3.250 ⋅ 6.247 = 20.30275
So, the margin of error is approximately 20.3 .
Calculate the Confidence Interval Finally, we calculate the 99% confidence interval for the true difference between the population means using the formula:
( d ˉ − E , d ˉ + E ) = ( − 4.1 − 20.3 , − 4.1 + 20.3 ) = ( − 24.4 , 16.2 )
Therefore, the 99% confidence interval for the true difference between the population means is ( − 24.4 , 16.2 ) .
State the Final Answer The 99% confidence interval for the true difference between the population means is ( − 24.4 , 16.2 ) .
Examples
Consider a scenario where a company implements a new training program and wants to assess its impact on employee performance. By comparing performance scores before and after the training for the same group of employees, they can use a paired difference test to determine if there's a significant change. The confidence interval provides a range within which the true average difference in performance lies, helping the company make informed decisions about the effectiveness of the training program. For example, if the confidence interval is (2, 15), it suggests the training likely improved performance. If the interval includes zero, like in our problem, the company can't be confident that the training had a significant impact.
The total charge delivered by a device with a current of 15.0 A over 30 seconds is 450 C. This translates to approximately 2.81 x 10^21 electrons flowing through the device. The calculations are based on the relationship between current, charge, and the charge of a single electron.
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